One obvious flaw I can point out is that Sony and Nintendo have drastically different images. To me, it'd appear to be mixing oil and water together.

1) Considering how the video games market works, it's not really an option to pull the plug on a console, if a company plans to stay in this business. By default, the new system would be surrounded by questions regarding its longevity. Consumers would be hesitant to invest into the hardware, if it could be discontinued within two years. There's no other option than to ride it out for four to five years. By the way, Iwata once said that Nintendo expected to sell about 50m GCs in its lifetime, so its safe to assume that the console was considered a failure. They also had the money to start over again quickly, but that isn't a good idea for the aforementioned reason.

2) Indeed, the Wii U was supposed to retain the Wii audience while winning back the hardcore (whoever those people are). It's a sort of strategy we've seen fail all too often when it comes to games. You know, when a game is supposed to be a sequel to a popular game, but isn't much like the previous game at all. The company in question is trying to take advantage of an established fanbase and sell to a bigger amount of people. Most of the time a big chunk of the fans is lost while the new people who the company wants to attract won't give the product a serious look, because they are put off by the name alone (they didn't care about the previous entry, so why should this one be any different). So at the moment the Wii U is pretty much selling only to the people who buy every Nintendo console. This can change with more and new games in the future, but the damage is already done.

3) The main difference between home consoles and handhelds is that Nintendo has rarely faced a competitor in the handheld market that was serious about it. Additionally, third parties are pretty much required to beat Nintendo, but since most third parties have never been serious about handhelds, Nintendo has never encountered as tough of a challenge in the handheld market as in the home console space. Hence why they came out on top in the handheld market every single time.

Meanwhile, in the home console market they have two big competitors nowadays. Companies that have been willing to lose billions on toys and courted third party developers to the best of their abilities (at least most of the time), to the point that third parties have come to expect incentives for developing for a platform. This basically leaves Nintendo with two options as a console manufacturer:

1. Make a system that plays to their own strengths and "sacrifice" the hardcore (whoever those people are). There's quite a big amount of people who can live without the so-called hardcore content. Nintendo has the IPs and talent to operate highly profitable with such a strategy, especially because most of the rest of the industry can't or doesn't want to sell to such people. It's going to secure Nintendo's spot in the long term.

2. Make a system that caters to third parties and thus becomes much like their competitors' consoles. Even if Nintendo succeeded at that, the profit margins would be thin, because the competition is so fierce.

One of those strategies is clearly better than the other, but for the eighth generation Nintendo went with the bad strategy, even though in the seventh generation they made it a clear point to avoid just that.

What exactly should the console have been like? Comparable processing power and RAM to other eighth generation consoles doesn't need explanation, but what kind of controller would be standard and how much would it cost at retail? Still though, if Sony got all the same third party games, I don't think Nintendo could have pushed Sony out of the market. That's because with all things being equal, people will choose the brand they trust. Hence why Microsoft couldn't beat Sony in Japan and Europe, despite having a few advantages.

Nintendo didn't follow the Wii formula, that's the entire problem. There are virtually no important similarities between the Wii and Wii U.

So how much would it cost? $350 without a game?

The main problem that remains though is that you assume Nintendo would have gotten every third party multiplatform game with such a console. Also, for its first year the console would have shared its multiplatform games with the 360 and PS3, thus many of the Wii U's current problems would exist in a similar way for this hypothetical Nintendo console. Expensive, not enough exclusive content, people holding out to see what Sony and Microsoft will bring to the table.